Singular PDEs

We propose in this work a theory of linear differential equations driven by unbounded operator-valued rough signals. As an application we consider rough linear transport equations and more general linear hyperbolic symmetric systems of equations driven by time-dependent vector fields which are only distributions in the time direction.

We provide in this work a semigroup approach to the study of singular PDEs, in the line of the paracontrolled approach developed recently by Gubinelli, Imkeller and Perkowski. Starting from a heat semigroup, we develop a functional calculus and introduce a paraproduct based on the semigroup, for which commutator estimates and Schauder estimates are proved, together with their paracontrolled extensions. This machinery allows us to investigate singular PDEs in potentially unbounded Riemannian manifolds under mild geometric conditions. As an illustration, we study the generalized parabolic Anderson model equation and prove, under mild geometric conditions, its well-posed character, in small time on a potentially unbounded 2-dimensional Riemannian manifold, for an equation driven by a coloured noise, and for all times for the linear parabolic Anderson model equation in 2-dimensional unbounded manifolds.

We sharpen in this work the tools of paracontrolled calculus in order to provide a complete analysis of the parabolic Anderson model equation in a 3-dimensional setting, in either bounded or unbounded domains equipped with a sub-Laplacian structure. We develop for that purpose a higher order paracontrolled calculus via semigroups methods. The technical core of this machinery is the introduction of a pair of intertwined space-time paraproducts on parabolic Holder spaces, with good continuity properties, as well as some continuity properties of iterated commutators and correctors built from paraproducts and resonant operators. Given the scope of our semigroup methods in terms of operators and geometry of the ambiant space, the application of our tools to the study of the 3-dimensional parabolic Anderson model equation provides results that go beyond the case of the 3-dimensional Euclidean space with its Laplace operator, very recently studied by Hairer and Labbe with the tools of regularity structures.

We develop in this work a general version of paracontrolled calculus that allows to treat analytically within this paradigm some singular partial differential equations with the same efficiency as regularity structures, with the benefit that there is no need to introduce the algebraic apparatus inherent to the latter theory. This work deals with the analytic side of the story and offers a toolkit for the study of such equations, under the form of a number of continuity results for some operators. We illustrate the efficiency of this elementary approach on the example of the generalised parabolic Anderson model equation in dimension 2+ and 3, and the generalized KPZ equation in an irregularity regime slightly better than the regime of space/time one dimensional white noise.

This work provides is a concise overview of paracontrolled calculus that appeared in the Séminaire des Journées EDPs.

We present in the first note a local in time well-posedness result for the singular 2-dimensional quasilinear generalized parabolic Anderson model equation, where the Laplacian in the equation is multiplied by a function of the unknown. The key idea of our approach is a simple transformation of the equation which allows to treat the problem as a semilinear problem and use the elementary setting of paracontrolled calculus, with no need of any extra ingredient.

An extension of the methods of high order paracontrolled calculus allows to use the approach implemented in the first note to study of a whole class of quasilinear singular PDEs, including a quasilinear version of the generalized (KPZ) equation.

We prove the well-posed character of a regularity structure formulation of the quasilinear generalized (KPZ) equation and give an explicit form for a renormalized equation in the full subcritical regime. Convergence results for the solution of the regularized renormalized equation are obtained in regimes that cover the spacetime white noise case.

We study the relation between the theory of regularity structures and paracontrolled calculus. We give in the first work a paracontrolled representation of the reconstruction operator and provide a natural parametrization of the space of admissible models. The second work provides a general equivalence statement between models and modelled distributions, and paracontrolled representations, and offers a number of illustrations of the main results. An overview is given in the following slides.

We give a short essentially self-contained treatment of the fundamental analytic and algebraic features of regularity structures and its applications to the study of singular PDEs.

Extended decorations on naturally decorated trees were introduced in the work of Bruned, Hairer and Zambotti on algebraic renormalization of regularity structures to provide a convenient framework for the renormalization of systems of singular stochastic PDEs within that setting. This non-dynamical feature of the trees complicated the analysis of the dynamical counterpart of the renormalization process. We provide a new proof of the renormalised system by-passing the use of extended decorations and working for a large class of renormalization maps, with the BPHZ renormalization as a special case. The proof reveals important algebraic properties connected to preparation maps.

Let T be the regularity structure associated with a given system of singular stochastic PDEs. The paracontrolled representation of the Π map provides a linear parametrization of the nonlinear space of admissible models over T, in terms of the family of para-remainders used in the representation. We give an explicit description of the action of the most general class of renormalization schemes on the parametrization space of the space of admissible models. The action is particularly simple for renormalization schemes associated with degree preserving preparation maps; the BHZ renormalization scheme has that property.

We develop in this note the tools of regularity structures to deal with singular stochastic PDEs that involve non-translation invariant differential operators. We describe in particular the renormalized equation for a very large class of spacetime dependent renormalization schemes.

We consider the continuous Anderson operator H on a two dimensional closed Riemannian manifold. We provide a short self-contained functional analysis construction of the operator as an unbounded operator on L2 and give almost sure spectral gap estimates under mild geometric assumptions on the Riemannian manifold. We prove a sharp Gaussian small time asymptotic for the heat kernel of H that leads amongst others to strong norm estimates for quasimodes. We introduce a new random field, called Anderson Gaussian free field, and prove that the law of its random partition function characterizes the law of the spectrum of H. We also give a simple and short construction of the polymer measure on path space and relate the Wick square of the Anderson Gaussian free field to the occupation measure of a Poisson process of loops of polymer paths. We further prove large deviation results for the polymer measure and its bridges.

We use some tools from nonlinear analysis to study two examples of singular stochastic elliptic PDEs that cannot be solved by the contraction principle or the Schauder fixed point theorem. Let H stand for the Anderson operator on a closed Riemannian surface. We prove the existence of a solution to the equation Hu = au + f(u), for a potentiel a in Lp and p between 1 and 2 and f subject to growth conditions. Under an additional parity condition on f we further prove that this equation has infinitely many solutions, in stark contrast with all the well-posedness results that have been proved so far for singular stochastic PDEs under a small parameter assumption. This kind of results is obtained by seeing the equation as characterizing the critical points of an energy functional and by resorting to variants of the mountain pass theorem. There are however some interesting equations that cannot be characterized as the critical points of an energy functional. Such is the case of the singular Choquard-Pecard equation. One can use Ghoussoub's machinery of self-dual functionals to prove the existence of a solution to that equation as the minimum of a self-dual strongly coercive functional under proper assumptions on the coefficients in the equation.

Rough paths

My first work on rough paths proposes an alternative approach to the theory of rough differential equations centred on the notion of flow of maps. By introducing a simple mechanism for constructing flows on a Banach space from approximate flows, one can reprove from scratch and extend in a simple way the main existence and well-posedness results for rough differential equations, in the context of dynamics on a Banach space driven by (potentially) (in)finite dimensional Holder weak geometric rough path; Taylor expansion and Euler estimates are also dealt with. This approach is illustrated by some existence and well-posedness results for some mean field stochastic, with mean field interaction in the drift only. As a direct application of the approach, we show in a note that it also works for equations driven by branched rough paths, and that Davie's notion of a solution to a rough differential equation and the notion of solution used here coincide.

We show in this note that the Ito-Lyons solution map associated to a rough differential equation is Frechet differentiable when understood as a map between some Banach spaces of controlled paths. This regularity result provides an elementary approach to Taylor-like expansions of Inahama-Kawabi type for solutions of rough differential equations depending on a small parameter, and makes the construction of some natural dynamics on the path space over any compact Riemannian manifold straightforward, giving back Driver's flow as a particular case.

The above work offers an extension to a non-semimartingale setting of the results on nonlinear differential equations and stochastic flows obtained in the 80's by Le Jan-Watanabe-Kunita and others, in the spirit of the method of approximate flows developped in my first work on rough paths theory. Here are some slides of a talk on the subject to get a quick overview of that work. Joint work with S. Riedel.

We introduce a notion of p-rough integrator on any Banach manifolds, for any p>1, which plays the role of weak geometric Holder p-rough paths in the usual Banach space setting. The awaited results on rough differential equations driven by such objects are proved, and a canonical representation is given if the manifold is equipped with a connection.

Together with J. Diehl, we tackled in the following work the elementary inverse problem. Is it possible to get back the driving rough path by observing the solution flow of a rough differential equation? The elementary example of an equation with constant vector fields makes it clear that this is not always possible. An elementary answer to the problem is given under the form of an algebraic rank condition on the driving vector fields and there brackets; it is unrelated however to Hormander's bracket condition.

We develop further in this work the theory of rough flows and show how it can be used to provide a conceptually simple approach to homogenization problems by working out in depth the case study of stochastic turbulence.

The present work aims at being a first step in linking the theory of random rough differential equations and random dynamical systems. We analyze here common lifts of stochastic processes to rough paths/rough drivers-valued processes and give sufficient conditions for the cocycle property to hold for these lifts. We show that random rough differential equations driven by such lifts induce random dynamical systems. In particular, our results imply that rough differential equations driven by the lift of fractional Brownian motion in the sense of Friz-Victoir induce random dynamical systems.

We give here a tight non-explosion criterion for solutions of rough differential equations driven by unbounded vector fields that provide an accurate picture of this question and corrects an imprecise statement of my work "Flows driven by rough paths".

We provide in this work a robust solution theory for random rough differential equations of mean field type, with mean field interaction in both the drift and diffusivity. Propagation of chaos results for large systems of interacting rough differential equations are obtained as a consequence, with explicit optimal convergence rate. The development of these results requires the introduction of a new rough path-like setting and an associated notion of controlled path. We use crucially Lions' approach to differential calculus on Wasserstein space along the way. Slides here.

We define in this note a notion of Young differential inclusion and give an existence result for such a differential system driven by mildy rough signals. As a by-product of our proof, we show that a bounded, compact-valued, a-Holder continuous set-valued map on the interval [0,1] has a selection with finite p-variation, for p>1/a. We also give a notion of solution to the rough differential inclusion driven by an a-Holder rough path, with a set-valued drift. We prove existence of a solution to the inclusion when F is bounded and lower semi-continuous with compact values, or upper semi-continuous with compact and convex values.

Diffusions on manifolds

We prove in the first work that bridges of subelliptic diffusions on a compact manifold, with different ends, satisfy a large deviation principle in the space of Holder continuous functions, with a good rate function, when the travel time tends to 0. This leads to the identification of the deterministic first order asymptotics of the distribution of the bridge under generic conditions on the endpoints of the bridge, in the sense that its law converges weakly to a Dirac mass on some particular deterministic path from between the two endpoints. It is natural in that setting to push further the analysis and try and get a second order asymptotics. We prove that the fluctuation process around the deterministic limit path is a Gaussian process whose covariance involves the (non-constant rank) sub-Riemannian geometry associated with the generator of the diffusion. This requires that the pair of end-points lies outside some intrinsic cutlocus associated with the generator. The following slides provide a gentle introduction to that work. The work on small time fluctuations for bridges of sub-Riemannian diffusions requires that we extend Varadhan's small time log-asymptotics for the heat kernel to possibly incomplete, unbounded, sub-Riemannian manifolds. This is the purpose of the third work.

In a joint work with J. Angst and C. Tardif, we introduce a Riemannian analogue of relativistic diffusions, called kinetic Brownian motion, whose flat version in a Euclidean space is a C1 random path run at constant speed κ, and whose velocity is a Brownian motion on the unit sphere run at constant speed σ. We show that, in both qualitive and quantitative way, that the trace in the manifold of these processes interpolate between geodesic and Brownian motions. We use rough paths theory to provide a simple and robust proof of this fact. We also investigate the long-time behaviour of these processes in rotationally invariant manifolds and determine their Poisson boundaries.

It is an interesting non-trivial task to define kinetic Brownian motion on the infinite dimensional 'manifold' of diffeomorphisms, or volume preserving diffeomorphisms, of a given compact manifold M. This construction is done in the following work, and we prove that kinetic Brownian motion provides an interpolation between the geodesic flow on the corresponding space and a Brownian flow on that space. In the Eulerian picture, this provides an intrinsic random perturbation of Euler's equations of incompressible fluids in the domain M. Slides here

Relativistic diffusions

In the same way as Brownian motion is the only continuous strong Markov process in the Euclidean space Rn whose law is invariant by the action of the isometries, there is essentially a unique way to define a C1 random path in the space of special relativity, representing the motion of an object having a speed less than the speed of light, and whose law is invariant by the action of the isometries of this space. The study of the asymptotic behaviour of this unique object, called "relativistic diffusion", is lead in these two articles using two different approaches. Whereas the first uses SDEs, stochastic calculus and couplings, the second (written with Albert Raugi) restates the problem in terms of the asymptotic behaviour of a random walk on a non semi-simple Lie group and highlights the deep geometric nature of the result.

Following the idea of C. Chevalier and F. Debbasch presented in their article 'Relativistic diffusions: a unifying approach', J. Math. Phys. 49 (4), 2008, we consider a class of relativistic diffusions, roughly characterized by the fact that there exists at each (proper) time (of the moving particle) a (local) rest frame where the acceleration of the particle is Brownian in any spacelike direction of the frame, when computed using the time of the rest frame. These diffusions are called by the authors relativistic diffusions. A pathwise approach of these processes is proposed here, in the general framework of Lorentzian geometry. The results proved not only provide a dynamical justification of the analytical approach developped up to now, and a new general H-theorem, it also sheds some light on the importance of the large scale structure of the manifold in the asymptotic behaviour of the Franchi-Le Jan process, studied in the above articles in Minkowski spacetime. See the following slides for a quick overview of that work, and the latter ones for some general presentation of the domain, with questions.

In so far as relativistic diffusions are defined in purely geometric terms, it is very likely that part (or all?) of the geometry of the ambient spacetime may be recovered from the probablistic behaviour of these processes. In a Riemannian setting, this probabilistic view on geometry is well-illustrated by Weyl and Pleyel formulas for the heat kernel of Brownian motion where local and global informations about the geometry appear. We investigate in this work one aspect of this geometry/probability correspondence. Dating back to Penrose and Hawking's results, it is now well-established that the appearance of singularities in Einstein's theory of gravitation is unavoidable under quite natural assumptions. Although there is no definitive agreement on what should be called a singularity of spacetime, a largely used notion of singularity is the existence in spacetime of incomplete geodesics. Is there a link between geodesic and probabilistic incompleteness? This work concentrates on one aspect of this question and provides different criteria under which the diffusion does not explode. A gentle introduction in the following slides .

The aim of this work is to promote the use of probabilistic methods in the study of problems in mathematical general relativity. Two new singularity theorems, whose features are different from the classical singularity theorems, are proved using probabilistic methods. Under some energy conditions, and without any causal or initial/boundary assumption, simple conditions on the energy flow imply probabilistic incompleteness. Also we introduce a probabilistic notion of spacetime boundary which has none of the pathological defects that the classical boundaries may have.

Following classical works initiated by Tanaka in the seventies in the framework of the space homogeneous Boltzmann equation, we show how one can associated intrinsically to the general relativistic version of Boltzmann equation a Markov process. It is used to prove the causal character of the general relativistic Boltzmann equation on strongly causal spacetimes.

Smoluchowski equation

Smoluchowski equation is a model of dynamics where clusters of different species coagulate at a rate depending on their characteristics. This model encompasses a large number of practical situations, from chemical reactions to powder storage... It is an important matter to understand how solutions of this equation depend on the parameters of the dynamics. The derivative of the solution with respect to the parameter is called the sensitivity. As a first step towards a detailled study of this dependence, I prove in this article that the measure-valued function representing the state of the system is a C1 function of these parameters, in a good space of measures.

The second article is the numerical counterpart of the above one. The theoretical result obtained there suggests a Marcus-Lushnikov like particle system to approximate the sensitivity. Convergence of the particle system is proved and numerical experiments are performed, showing the great accuracy and low variance of the estimator.

The third article presents an alternative method to simulate the sensitivity in which a coupled pair of Marcus-Lushnikov process is used to simulate two solutions to Smoluchowski equation with close parameters.

I prove that the spatial coagulation equation with bounded coagulation rate is well-posed for all times in a given class of kernels if the convection term of the underlying particle dynamics has divergence bounded above by a negative constant. Multiple coagulations, fragmentation and scattering are also considered.


Exponential functional of Brownian motion. In this note, I give a short proof of a result of Dufresne on some exponential functional of a geometric Brownian motion.

KPZ identity. This note (not to be published) shows how to extend Benjamini and Schramm's one dimensional KPZ identity in a mutlidimensional random geometry of multiplicative cascades.

Habilitation work.

PhD work. My PhD work dealt with relativistic diffusions, especially finding the Poisson boundary of Dudley's diffusion in Minkowski spacetime.